Received: 7 May 2009 – Revised: 27 July 2009 – Accepted: 27 July 2009 – Published: 5 August 2009
Abstract. Aliasing is a general problem in the analysis ofany measurements that make sampling at discrete points.Sampling in the spatial domain results in a periodic patternof spectra in the wave vector domain. This effect is calledspatial aliasing, and it is of particular importance for multi-spacecraft measurements in space. We first present the theo-retical background of aliasing problems in the frequency do-main and generalize it to the wave vector domain, and thenpresent model calculations of spatial aliasing. The model cal-culations are performed for various configurations of the re-ciprocal vectors and energy spectra or distribution that areplaced at different positions in the wave vector domain, andexhibit two effects on aliasing. One is weak aliasing, inwhich the true spectrum is distorted because of non-uniformaliasing contributions in the Brillouin zone. It is demon-strated that the energy distribution becomes elongated in theshortest reciprocal lattice vector direction in the wave vec-tor domain. The other effect is strong aliasing, in whichaliases have a significant contribution in the Brillouin zoneand the energy distribution shows a false peak. These resultsgive a caveat in multi-spacecraft data analysis in that spectralanisotropy obtained by a measurement has in general two ori-gins: (1) natural and physical origins like anisotropy imposedby a mean magnetic field or a flow direction; and (2) aliasingeffects that are imposed by the configuration of the measure-ment array (or the set of reciprocal vectors). This manuscriptalso discusses a possible method to estimate aliasing contri-butions in the Brillouin zone based on the measured spectrumand to correct the spectra for aliasing.
Keywords. Electromagnetics (Antenna arrays; Signal pro-cessing and adaptive antennas) – Space plasma physics (Ex-perimental and mathematical techniques)
Recent progress in our understanding of near-Earth spaceenvironments such as the solar wind or the Earth’s mag-netosphere benefits greatly from multi-spacecraft missionsthat enable us to distinguish between temporal and spatialvariations in the observations, and allows us to determinethree-dimensional structures in space. The Cluster mission(Escoubet et al., 2001; Balogh et al., 2001) performs four-point measurements with its tetrahedron configuration andthe THEMIS mission (Angelopoulos, 2008; Auster et al.,2008) performs five point measurements. There exist a num-ber of data analysis methods that are developed particularlyfor such multi-point measurements, parts of which are sum-marized in Glassmeier et al.(1995) and Paschmann andDaly (1998, 2008). One of the unique and powerful anal-ysis methods in studying dynamics of space plasma is todetermine the energy distribution directly in the wave vec-tor domain, referred to as the wave telescope technique (ork-filtering) (Neubauer and Glassmeier, 1990; Pincon andLefeuvre, 1991; Motschmann et al., 1996; Glassmeier et al.,2001). This technique is based on the minimum variancemethod ofCapon(1969) estimating the energy spectrum ordistribution in the frequency and wave vector domain usingonly several points of measurements.
Energy distributions for fluctuating magnetic field in thewave vector domain have been determined using Clusterand have been presented by several authors such asGlass-meier et al.(2001); Narita et al. (2006, 2007); Sahraouiet al. (2006). Dispersion relations of plasma normal modescan also be determined experimentally from the Cluster data(Narita et al., 2003; Narita and Glassmeier, 2005; Vogt,2008). Figure 1 displays an example of the energy spec-trum in the wave vector domain, observed by Cluster space-craft in the shock-upstream region. The fluctuation energyis anisotropically distributed between two directions: awayfrom the shock toward the sun (E+ in the spectrum), and
Published by Copernicus Publications on behalf of the European Geosciences Union.
3032 Y. Narita and K.-H. Glassmeier: Spatial aliasing
Fig. 1. Example of anisotropic energy spectrum as a function ofthe wave number parallel to the mean magnetic field, taken fromthe Cluster observation in the shock-upstream region afterNaritaet al.(2007). The solid curve represents the spectrum for magneticfield fluctuations in the direction away from the bow shock (E+)and the dotted curve represents that in the direction toward the bowshock (E−).
from the sun toward the shock (E−). While the larger energycontribution inE+ can be physically interpreted as the ex-citation of the waves propagating away from the bow shock,one may also ask how important the aliasing effect is in eval-uating the energy spectrum.
It is worthwhile to note that spatial structures can not al-ways be resolved and identified in the multi-point measure-ments. For example, one can not identify the spatial struc-ture when its characteristic scale (e.g. wavelength) is smallerthan the sensor separation distance. In that case the measure-ment makes under-sampling and the result is subject to alias-ing problem.Neubauer and Glassmeier(1990) presented theconcept of spatial aliasing under multi-point measurementsin space, and measurements using the four Cluster spacecraftindeed suggest the existence of spatial aliasing (Sahraouiet al., 2003; Tjulin et al., 2005).
The configuration of the sensor array can be representedas a set of reciprocal vectors in the wave vector domain, andthe analysis in the wave vector domain can be performedwithin the first Brillouin zone or Wigner-Seitz cell (hereafterreferred to as the Brillouin zone or simply the cell) spannedby the half-length reciprocal vectors (Kittel, 1996). Beyondthe Brillouin zone the determined spatial structures do notreflect the true structure any more, instead its aliases appearsperiodically in the wave vector domain.
Aliasing is a general problem not only in science but alsoin engineering so far as discrete measurements sample a con-tinuous function. The goals of this paper are (1) to general-ize aliasing theory from the frequency domain to the wavevector domain, and (2) demonstrate various cases of spatialaliasing in the wave vector domain. It is concluded that the
configuration of sensor arrays may significantly influence themeasurement of spatial structures due to the aliasing problemeven though the analysis is limited within the Brillouin zone.We also discuss a remedy to correct for the aliasing part inthe measurement.
2 Theoretical background
2.1 Sampling formulas in time domain
Sampling formulas, their mathematical foundation, and thealiasing problem have been developed and studied in a num-ber of papers, not only in the celebrated work ofNyquist(1928) and Shannon(1949) but also in recent papers suchasUnser(2000) andKirchner(2005). We briefly discuss therelation between sampling in the time domain and the associ-ated aliasing in the frequency domain. This is essentially analiasing problem in a one-dimensional system and providesa basis for understanding the aliasing in the wave vector do-main.
Spectral aliasing in some under-sampled time series can beunderstood in the following fashion. Consider a continuousfunction g(t) in the time domain and its Fourier transformG(f ) in the frequency domain. The functiong(t) is mea-sured at discrete points, yielding our sampled signalh(t)
h(t) = g(t)III (t), (1)
where III(t) is called the comb distribution (Bracewell, 2000)which represents an array of delta functions
III (t) =
∞∑m=−∞
δ(f0t −m). (2)
Here f0 denotes the sampling ratef0 = 1/1t . The combdistribution is a periodic function and can be expressed as aFourier series
III (t) =
∞∑n=−∞
cnei2πnf0t . (3)
Furthermore, it turns out that the Fourier coefficientscn areall unity:
cn =1
1t
∫ 1t2
−1t2
δ(f0t)e−i2πnf0tdt = 1, (4)
where one of the properties of the delta function,δ(f0t) =
δ(t)/f0 was used. The Fourier transform of the sampled sig-nalh(t) is obtained by combining Eqs. (1), (3), and (4).
H(f ) =
∫∞
−∞
h(t)e−i2πf0tdt (5)
=
∫∞
−∞
g(t)III (t)e−i2πf0tdt (6)
=
∞∑n=−∞
∫∞
−∞
g(t)e−i2π(f −nf0)tdt (7)
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Y. Narita and K.-H. Glassmeier: Spatial aliasing 3033
=
∞∑n=−∞
G(f −nf0). (8)
In the last equation one sees the essence of the aliasing prob-lem, that is the Fourier transform of the sampled signal con-tains not only the Fourier transform of the original func-tion G(f ) but also contributions from then-th harmonics ofthe sampling frequencyG(f −nf0). The power spectrum isproportional to the expectation value of the squared Fouriertransform. On the assumption that the phase between anytwo frequency components is random, the power spectrum isalso a sum of the respective alias component,
P(f ) = E[|H(f )|2
]=
∞∑n=−∞
|G(f −nf0)|2, (9)
where the symbolE[···] denotes the operation of the expec-tation value. Figure2 displays the periodic pattern of powerspectrum. In this example the contribution from aliases issignificant and the true spectrum (top panel) becomes en-hanced and distorted in the aliased spectrum (bottom panel)that takes account of the true spectrum and its alias contribu-tions, even within the Nyquist frequencyfN = f0/2.
2.2 Sampling formulas in spatial domain
The above sampling formulas can be generalized to the spa-tial domain. Consider now a continuous function of vector inreal space,g(r), and its Fourier transform in the wave vec-tor domain,G(k). The sampled signal is a filtering of thefunctiong(r) by the set of the delta functions III(r), namely
h(r) = g(r)III (r). (10)
The comb distribution in the spatial domain is expressed as
III (r) =
∞∑m3=−∞
∞∑m2=−∞
∞∑m1=−∞
δ
(k3 · r2π
−m3
)δ
(k2 · r2π
−m2
)δ
(k1 · r2π
−m1
)(11)
=
∞∑m3=−∞
δ
(k3 · r2π
−m3
) ∞∑m2=−∞
δ
(k2 · r2π
−m2
)∞∑
m1=−∞
δ
(k1 · r2π
−m1
)(12)
or as a Fourier series
III (r) =
∞∑n3=−∞
ein3k3·r∞∑
n2=−∞
ein2k2·r∞∑
n1=−∞
ein1k1·r (13)
=
∞∑n3=−∞
∞∑n2=−∞
∞∑n1=−∞
ei(n1k1+n2k2+n3k3)·r , (14)
wherek1, k2, andk3 are the reciprocal vectors of the primi-tive vectors of the sensor array (or spacecraft separation vec-tors)r1, r2, andr3. Equation (12) is a grouping of the sums
Fig. 2. Illustration of aliasing in the frequency domain. A truespectrum (top panel) appears periodically as aliases in the frequencydomain with the frequency shiftnf0 (middle panel). This results indistortion of the original spectrum (bottom panel) even within theNyquist frequencyfN (shaded area).
in Eq. (11) with respect tom1, m2, andm3. Equation (13) isobtained by Fourier series expansion of the three delta func-tions in Eq. (12) (cf. Eqs. 2–4). Equation (14) combines threeexponential functions in Eq. (13) together.
The reciprocal vectors are obtained by the primitive vec-torsr1, r2, andr3 as
k1 = 2πr2×r3
r1 ·(r2×r3)(15)
k2 = 2πr3×r1
r2 ·(r3×r1)(16)
k3 = 2πr1×r2
r3 ·(r1×r2), (17)
satisfying the orthogonality condition:
ki ·rj = 2πδij , (18)
whereδij represents Kronecker’s delta. The set of these threereciprocal vectors used in this study are the canonical choicein solid state physics. An alternative set of reciprocal vectorsis also in use (Chanteur, 1998). The Fourier transform ofh(r) is given as
H(k) =
∫h(r)e−ik·rd3r (19)
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3034 Y. Narita and K.-H. Glassmeier: Spatial aliasing
Fig. 3. Aliasing in the wave vector domain. The shaded disks rep-resent the energy distribution with its aliases. The hatched area withdash-lines is the first Brillouin zone (Wigner-Seitz cell) spanned bythe half-length reciprocal vectors.
=
∑n1,n2,n3
∫g(r)e−i(k−n1k1−n2k2−n3k3)·rd3r (20)
=
∑n1,n2,n3
G(k−n1k1−n2k2−n3k3) (21)
Again on the assumption of random phase, one obtains thespectrum in the wave vector domain as
P(k) = E[|H(k)|2
](22)
=
∑n1,n2,n3
|G(k−n1k1−n2k2−n3k3)|2 (23)
Therefore the spatial sampling results in a periodic struc-ture of the spectrum in the wave vector domain, containingboth the true spectrum|G(k)|2 and its aliases|G(k−n1k1−
n2k2−n3k3)|2. Figure3 displays an illustration of the spec-
trum with aliasing in the wave vector domain.As Neubauer and Glassmeier(1990) state, the periodic
structure in the wave vector domain is avoided if we restrictthe analysis to a small subvolume of the wave vector space.For example, one can use a sphere in the wave vector do-main with the radius equal to the minimum reciprocal vectorlength, or one can use a right square or cube with this length.It is important that the analysis should be limited to withinthe Brillouin zone, otherwise the spectrum exhibits a peri-odic alias pattern in the wave vector domain. However, westress here that the original spectrum may be enhanced anddistorted as shown in Fig.2 even if we restrict ourselves tothe Brillouin zone.
3 Aliasing in the wave vector domain
3.1 An example
Figure 4 displays an example of aliasing in a two-dimensional wave vector domain. The data (the true spec-trum) are generated numerically and the associated aliasingeffect is studied. The wave vector domain is spanned bythe half-length reciprocal vectorsk1 = (0.02,0.00) km−1 andk2 = (0.00,0.01) km−1. The true spectrum (left panel) repre-sents an isotropic energy distribution, centered almost at theorigin (kx,ky) = (0.0001,0.000) km−1 and decaying towardlarger wave numbers by a power-lawP(|k|) ∝ k−α with theindex α = 5/3. Contribution from the nearest neighboringaliases with(n1,n2) = (±1,±1), (0,±1), (±1,0) in the firstBrillouin zone is displayed in the middle panel. A large partof the alias contribution comes from the alias(n1,n2) = (0,1)
and (0,−1), since the magnitude of the second reciprocalvectork2 is smaller than that of the first one and the aliaspartnersG(k −n1k1 −n2k2) are spaced in the wave vectordomain more closely to each other in the direction tok2. Theright panel in Fig.4 displays the aliased spectrum which isa sum of the true spectrum (left panel) and the aliased part(middle panel). The aliased spectrum is what we expect toobtain from multi-point measurements. It is interesting thatthe aliased spectrum does not present an isotropic distribu-tion any more, but is elongated in the direction tok2 whichis the shortest lattice vector direction in the reciprocal space.It is clear at this stage that the elongation or anisotropy in theenergy spectrum or distribution is imposed by the the con-figuration of the reciprocal vectors, especially how close ordifferent the magnitudes of the vectors|k1| and|k2| are fromeach other. Another effect is that the spectrum is enhancedat various positions in the Brillouin zone. On the other hand,the isotropy in the spectrum is only slightly broken aroundthe peak. The reason for this is that the spectral energy islarger near the peak and furthermore it is at enough distancefrom the border of the Brillouin zone such that the contribu-tion from the aliases is the smallest there.
Distortion of the spectrum due to aliasing is investigatedusing two parameters: an enhancement parameterE and ananisotropy parameterA. The enhancement parameter is de-fined as a ratio of the aliased spectrumPals to the true spec-trum Ptrue as a function of wave number from the spectralpeak in a given direction,
E(k) =Pals(k)
Ptrue(k). (24)
We choose the maximum and the minimum enhancementdirection in this study. Figure5 displays the enhancementparameter in the example case in these two directions (de-noted as “max” and “min” in the aliased spectrum). Inthe maximum enhancement direction this quantity increasesfrom E=1.00 at the spectral peak (which isk=0.0000 km−1
in Fig. 5) to E=2.51 at the largest deviation wave number
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Y. Narita and K.-H. Glassmeier: Spatial aliasing 3035
Fig. 4. True spectrum (left), aliased part from the neighboring cells (middle), and aliased spectrum that is a sum of the true spectrum and itsaliases (right).
Table 1. Enhancement parameter and anisotropy parameter in theexample case.
k (km−1) 0.0000 0.0020 0.0040
max(E) 1.00 1.26 1.99min(E) 1.00 1.25 1.77
A(%) 0 2 12
k=0.0048 km−1. In the minimum enhancement direction italso increases but more moderately than in the maximum di-rection,E=1.00 at the spectral peak,E=2.02 at 0.0048 km−1,andE=4.34 at 0.095 km−1. Some more values of the en-hancement parameter are listed in Table 1 and plotted inFig. 5.
We then investigate the aliased spectrum with theanisotropy parameter that measures a relative difference inthe aliased spectrum between the maximum and the mini-mum enhancement directions:
A =max(Pals)−min(Pals)
min(Pals). (25)
Alternatively one may use the enhancement parameterE =
Pals/Ptrue instead of the aliased spectrumPals if the true spec-trum is isotropic, i.e.,Ptrue is independent from the choice ofdirection from the spectral peak and only a function of themagnitude of the wave number. We evaluate the anisotropyparameter at two different wave numbers (Table 1 bottom).Anisotropy increases as the wave number becomes larger, upto A=24% at the largest wave number available in the max-imum enhancement direction, which isk=0.0048 km−1 (atthe right edge of the “max” curve in Fig.5).
Fig. 5. Enhancement parameter (ratio of aliased to true spectrum)in the maximum (dotted line) and minimum distortion directions(solid line) for the spectrum in Fig.4. The x-axis measures distancefrom the peak of the spectrum in the wave number domain.
3.2 Effect of spectral peak positions and reciprocalvectors
Here we study how aliasing affects the true spectrum underdifferent conditions: (case 1) the peak position is shifted inthe true spectrum; (case 2) the lengths of the reciprocal vec-tors are changed for the rectangular Brillouin zone; (case 3)the shape of the Brillouin zone is changed from a rectangularto a diamond-shaped cell; and (case 4) the peak position isshifted in the diamond-shaped cell.
Case 1: Peak position
The effect of the peak position in the Brillouin zone is firstinvestigated. Figure6exhibits three cases of the aliased spec-trum for the same Brillouin zone configuration as the ex-ample case in Fig.4. The true spectrum is again the sameisotropic, power-law distribution but its peak is displaced todifferent positions in the Brillouin zone, near the origin at
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3036 Y. Narita and K.-H. Glassmeier: Spatial aliasing
Fig. 6. Aliased spectra in the wave vector domain including both the true and the aliases spectrum for different peak positions of the truespectrum shifted in the Brillouin zone. Peak near the origin (left), at an intermediate distance to the border of the Brillouin zone (middle),near the border (right). Reciprocal vectors are orthogonal to each other and the Brillouin zone forms a rectangular cell.
kc = (0.0015,0.0015) km−1 in case (1a); at an intermediatedistance from the origin and the border of the Brillouin zone,(0.0030,0.0030) km−1, in case (1b); and almost at the borderat (0.0045,0.0045) km−1 in case (1c).
In all three cases the distribution can be divided intotwo parts: the distribution is isotropic near the peak, andanisotropic in its surrounding in the sense that the contourlines are elongated and aligned with the shortest lattice vec-tor direction, parallel tok2. In case (1a), the distributionexhibits a distortion in the lower half plane in the Bril-louin zone, a neck-shaped contour line, due to the alias from(n1,n2) = (0,−1). In case (1b), the peak is shifted closerto the upper border of the Brillouin zone. The top part ofthe true spectrum is not measurable any more as it is beyondthe Brillouin zone. Instead its alias(n1,n2) = (0,−1) entersthe Brillouin zone. This part of the spectrum corresponds tothe missing part of the spectrum at the top of the Brillouinzone. In case (1c), the peak is again shifted even closer to theborder of the Brillouin zone. The distribution exhibits nowtwo peaks, one at the top of the cell (which reflects the orig-inal spectrum) and the other at the bottom which is the alias(n1,n2) = (0,−1). The transition from the case (1a) to (1c)suggests that the true spectrum is distorted and elongated un-der a moderate alias contribution, while a false peak appearsfrom the opposite side of the Brillouin zone under strongaliasing. The elongation is roughly aligned with the short-est lattice vector direction, and the distortion of the spectrumis largest where the aliasing contribution is largest in the Bril-louin zone.
Case 2: Rectangular cell shape
Not only the peak position but also the shape of the Bril-louin zone is important in the spatial aliasing. Figure7compares three different rectangular Brillouin zones and thealiased spectra, while the true spectrum is the same in case
case, centered atkc = (0.0005,0.0005) km−1 with the sameisotropic, power law distribution as the former examples.The case (2a) exhibits the Brillouin zone spanned by thehalf-length of reciprocal vectorsk1 = (0.020,0.000) km−1
andk2 = (0.000,0.016) km−1, where the magnitude of thetwo reciprocal vector is not much different from each other.The measured distribution is almost isotropic. The distor-tion or elongation of the spectrum can be identified only nearthe border of the Brillouin zone. The case (2b) exhibits thesecond reciprocal vector with half-length of the first one,k1 = (0.020,0.000) km−1 and k2 = (0.000,0.010) km−1.Anisotropy in the aliased spectrum becomes clearer and thedistribution is elongated to the shortest lattice vector, in thedirection tok2. The isotropic distribution can also be identi-fied but its area in the Brillouin zone is smaller than case (2a).The second reciprocal vector is more shortened in case (2c),with k1 = (0.020,0.000) km−1 andk2 = (0.000,0.04) km−1.The anisotropy is even stronger in this case and the contourlines are almost aligned with thek2 direction. The isotropicdistribution can be identified only near the peak.
Case 3: Diamond-shaped cell
Reciprocal vectors are not always mutually orthogonal buthave in general various angles. In such a case the Brillouinzone forms a diamond-shaped cell in the wave vector do-main. Figure8 compares the Brillouin zones with three dif-ferent angles of the reciprocal vectors: 50◦ (3a), 30◦ (3b),and 20◦ (3c). Again, the true spectrum is the same as the rect-angular cell case in Fig.7, isotropic, power law distributionwith α=5/3 centered atkc = (0.0005,0.0005) km−1. The re-ciprocal vectors in case (3a)–(3c) have all the same magni-tude. In case (3a) the Brillouin zone is spanned by the half-length of the reciprocal vectorsk1 = (0.0147,0.0053) km−1
an k2 = (0.0053,0.0147) km−1 with the mutual angle 50◦.The distribution is only moderately distorted and almost
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Y. Narita and K.-H. Glassmeier: Spatial aliasing 3037
Fig. 7. Aliased spectra for different rectangular cell shapes of the Brillouin zone, close to right square (left), moderately narrow in thevertical reciprocal vector length (middle), even narrower in that length (right). The dotted lines are parallel to the reciprocal vectors.
Fig. 8. Aliased spectra for different diamond-shaped Brillouin zones, with the angle between the two reciprocal vectors 50◦ (left), 30◦
(middle), and 20◦ (right).
isotropic except for the region near the corner of the Brillouinzone. The distribution is weakly elongated in the directionto the left-top and right-bottom corners which represents theshortest lattice vector direction,−k1 +k2 (toward the left-top corner of the cell) or−k1+k2 (toward the right-bottomcorner). Case (3b) exhibits the mutual angle 30◦ betweenthe two reciprocal vectors,k1 = (0.0127,0.0073) km−1 ank2 = (0.0073,0.0127) km−1. The area of the isotropic distri-bution becomes smaller and the contour line are more alignedwith the shortest lattice vector direction. Aliasing contribu-tion can also be found near the right-top and the left-bottomcorners (in the direction to longest lattice vector directions,k1 + k2) and the spectral energy increases near these cor-ners. Case (3c) exhibits even narrower angle, 20◦ with k1 =
(0.0118,0.0082) km−1 and k2 = (0.0082,0.0118) km−1.The isotropic distribution area is again small and the distribu-tion is overall anisotropic, though the elongation of the dis-tribution is not very much different from case (3b). At theright-top and the left-bottom corners the distribution exhibits
an enhanced contribution from the aliases such that the spec-tral energy first decreases and then increases near the cornerof the cell in the longest lattice vector directions.
Case 4: Peak position in diamond-shaped cell
Peak position of the true spectrum is shifted in the diamond-shaped Brillouin zone with the angle 30◦, the same con-figuration as Fig. 8 middle panel. The shift direc-tion is toward the upper border of the cell. Figure9compares the peak position shifted to the upper side ofthe Brillouin zone, fromkc = (0.0005,0.0010) km−1 incase (4a), to(0.0005,0.0020) km−1 in case (4b), and thento (0.0005,0.0030) km−1 in case (4c). In case (4a) theisotropic distribution can be seen only near the peak, oth-erwise the overall distribution is moderately anisotropic andelongated to the shortest lattice vector direction (−k1+k2).In case (4b) the distribution has again a small isotropic regionnear the peak and an elongated structure in its surrounding.
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3038 Y. Narita and K.-H. Glassmeier: Spatial aliasing
Fig. 9. Aliased spectra for different peak positions in a diamond-shaped Brillouin zone. The peak is shifted to the top side of the Brillouinzone. Peak close to the origin in the cell (left), between the origin and the border of the cell (middle), and near the border (right).
Fig. 10. Enhancement parameter in the maximum (dotted lines) and the minimum distortion directions (solid lines) for the twelve spectrapresented in Figs.6 to 9.
There is an asymmetry in the distribution between the right-top and left-bottom corners. The distorted distribution at theleft-bottom corner comes from the alias(n1,n2) = (0,−1).In case (4c) this alias enters more in the Brillouin zone andtwo populations are identified in the distribution, the origi-nal peak at the top of the cell and its alias at the left-bottomcorner. The distribution is isotropic near these peaks, oth-erwise the contour lines are elongated to the left-top cornerdirection.
3.3 Quantitative analysis
The enhancement parameterE and the anisotropy parame-ter A are determined for the twelve spectra (1a) to (4c) to
quantify the effect of aliasing. The enhancement parameterin each case is summarized in Table 2 at three distinct wavenumbers from the spectral peak, and is also plotted in Fig.10.
Case 1
All the three spectra (1a), (1b), and (1c) exhibit very similarenhancement: an increase fromE = 1.00 at the peak (k =
0.0000 km−1) to E = 3.85 atk = 0.0060 km−1 in the maxi-mum direction, and also an increase from fromE = 1.00 toE = 2.49 in the minimum direction. The anisotropy param-eter increases fromA = 0% at 0.0020 km−1 to A = 12% at0.0040 km−1 and toA = 55% at 0.0060 km−1 in the threecases. The anisotropy parameter at 0.0060 km−1 is plotted
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Y. Narita and K.-H. Glassmeier: Spatial aliasing 3039
Table 2. Enhancement parameter for the twelve spectra in Figs.6to 9 in the maximum and the minimum distortion directions.
k (km−1) 0.002 0.004 0.006
case 1a max(E) 1.26 1.99 3.85min(E) 1.25 1.77 2.49
1b max(E) 1.26 1.99 3.85min(E) 1.25 1.77 2.49
1c max(E) 1.26 1.99 3.85min(E) 1.25 1.77 2.49
case 2a max(E) 1.16 1.54 2.15min(E) 1.16 1.53 2.06
2b max(E) 1.26 1.99 –min(E) 1.25 1.77 2.49
2c max(E) 2.29 – –min(E) 1.65 2.56 3.43
case 3a max(E) 1.24 1.82 2.74min(E) 1.24 1.80 2.66
3b max(E) 1.43 – –min(E) 1.38 2.18 3.35
3c max(E) 1.81 – –min(E) 1.57 2.56 3.84
case 4a max(E) 1.43 – –min(E) 1.38 2.18 3.35
4b max(E) 1.43 2.86 –min(E) 1.38 2.18 3.35
4c max(E) 1.43 2.86 –min(E) 1.38 2.18 3.35
in Fig. 11 (left top) as a function ofκ which is a relativecloseness of the spectral peak to the boundary of the Bril-louin zone,
κ =
∣∣∣∣kc −kb
kb
∣∣∣∣. (26)
Herekb denotes the boundary wave vector of the Brillouinzone closest from the spectral peakkc.
Case 2
In case (2a) the enhancement parameter almost degener-ates between the maximum and the minimum directions,and it increases fromE = 1.00 at the peak toE = 2.15(maximum) andE = 2.06 (minimum) at the wave number0.0060 km−1. Degenerated enhancement curves are a signof almost isotropic distribution, and in fact the anisotropyparameter isA = 4% at 0.0060 km−1. In case (2b) the max-imum enhancement increases fromE = 1.00 at the spectralpeak toE = 2.7 at 0.0050 km−1. The largest wave num-ber in the maximum direction is diminished, reflecting thefact that the second reciprocal vectork2 is shorter than thatin case (2a). The minimum enhancement in (2b) increasesfrom E = 1.00 at the peak toE = 2.49 at 0.0060 km−1.In case (2c) the wave number range for the maximum en-hancement is even shorter, and varies fromE = 1.00 at the
Fig. 11. Anisotropy parameter as a function of the closenessκ ofthe spectral peak to the boundary of the Brillouin zone measured atk = 0.0060 km−1 from the spectral peak (case 1), the ratio of thetwo reciprocal vector magnitudes atk = 0.0024 km−1 (case 2), theangle between the two reciprocal vectors atk = 0.0022 km−1 (case3), and the peak-boundary closenessκ atk = 0.0036 km−1 (case 4).
peak toE = 3.33 at 0.0024 km−1, whereas the minimum en-hancement varies fromE = 1.00 at the peak toE = 1.84at 0.0024 km−1 and to E = 3.43 at 0.0060 km−1. Theanisotropy parameter measured atk = 0.0020 km−1 is A =
0% (2a), 1% (2b), and 38% (2c). The anisotropy parame-ter increases more rapidly from (2a) to (2c) when measuredat a larger wave number. Figure11 displays the anisotropyparameter 4% (2a) to 29% (2b) and to 81% (2c) measuredat k = 0.0024 km−1 (the maximum wave number availablein case (2c)) as a function of the ratio of the two reciprocalvectors|k2/k1|. Anisotropy becomes stronger for a shortersecond reciprocal vectork2.
Case 3
In case (3a) the enhancement again almost degeneratesand varies from E = 1.00 at the spectral peakk =
0.0000 km−1 to E = 1.24 (both maximum and minimum) atk = 0.0020 km−1, and toE = 2.74 (maximum) andE = 2.66(minimum) at k = 0.0060 km−1. In case (3b) the max-imum enhancement varies fromE = 1.00 to E = 1.43 atk = 0.0020 km−1, and toE = 2.24 atk = 0.0034 km−1. Theminimum enhancement varies fromE = 1.00 toE = 1.38 at0.0020 km−1, and toE = 1.90 atk = 0.0034 km−1. In case(3c) the maximum enhancement varies fromE = 1.00 toE =
1.81 at 0.0020 km−1. The minimum enhancement varies
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3040 Y. Narita and K.-H. Glassmeier: Spatial aliasing
Fig. 12. Reconstruction of the spectrum by correcting the aliases. The spectrum is extrapolated for the neighboring cells in the wavevector domain (left panel), and then the alias of the extrapolated parts are estimated in the Brillouin zone (middle panel). The corrected orreconstructed spectrum is displayed in the right panel.
from E = 1.00 toE = 1.57 at 0.0020 km−1. The anisotropyparameter increases moderately toward larger wave numbersin each case. For example, case (3a) exhibits the anisotropyparameterA = 0%, 1%, and to 3% atk = 0.0020 km−1,0.0040 km−1, and 0.0060 km−1, respectively. Anisotropy in-creases a little more when the angle between the two recipro-cal vectors becomes narrower when measured at a fixed wavenumber. The anisotropy parameter is plotted as a functionof the angle between the two reciprocal vectors in Fig.11.Anisotropy increases for smaller angles of reciprocal vec-tors, and it varies from 0% (3a) to 4% (3b) and to 18% (3c)atk = 0.0022 km−1 (largest wave number available in (3c)).
Case 4
Wave number range in the maximum enhancement direc-tion is short in (4a) and largest in (4c), but the three casesdo not exhibit any significant difference in the enhance-ment parameter. The maximum enhancement parameter in-creases fromE = 1.00 at the spectral peak toE = 1.43 atk = 0.0020 km−1 in (4a), (4b), and (4c), and then toE = 2.86at k = 0.0040 km−1 in (4b) and (4c). The enhancement pa-rameter in the minimum direction increases fromE = 1.00at the peak toE = 1.38 at k = 0.0020 km−1, E = 2.18 atk = 0.0040 km−1, andE = 3.35 atk = 0.0060 km−1 in (4a)to (4c). The anisotropy parameter isA = 4% in the threecases when measured at= 0.0020 km−1. Figure11plots theanisotropy parameter measured at= 0.0036 km−1 (largestwave number available in (4a)) as a function ofκ, the close-ness of the spectral peak to the boundary of the Brillouinzone.
4 Correction of spatial aliasing
The procedure of correcting the spectrum for aliasing effectis mathematically to estimate in Eq. (9) or (23) the func-tion G for given functionP . There is, however, an uncer-tainty in this inverted-procedure because the true spectrumG must first be known to estimate the contribution of thealiases. The correction can in principle be achieved by aniteration scheme, that is to guess a true spectrum and then es-timate aliased spectrum using the reciprocal vectors and theassumed true spectrum. If the guessed spectrum yields analiased spectrum which is close to the measured one, thenthe guessed spectrum can be used to correct the spectrumby subtracting the aliased part from the measured spectrum.If the guessed spectrum yields a different aliased spectrumfrom the measured one, then we modify the guessed spec-trum and repeat fitting procedure of the measured spectrumuntil the guessed spectrum provides a reasonably close spec-trum to the measurement.
One of the ways to guess a true spectrum is to investi-gate the measured spectrum near its peak. As the quantita-tive analysis suggests, aliasing effect is smallest in this re-gion and the spectrum near the peak can potentially be usedfor an extrapolation of the energy distribution in the area ex-terior of the Brillouin zone. Aliased spectrum is then es-timated based on the extrapolated spectrum, and the mea-sured spectrum is corrected for the aliases. Figure12 em-ploys this concept and applies the correction method to theexample case of Fig.4. The extrapolation is performed forthe core distribution in the neighboring cells of the Brillouinzone (Fig.12 left), where the peak and the power-law in-dex for the core distribution are determined in the measureddistribution by a fitting procedure. The extrapolated distribu-tion is displaced to the neighboring reciprocal lattice points(aliasing atn1 = ±1,n2 = ±1), and the total contributionfrom these aliases is estimated in the Brillouin zone (Fig.12
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Y. Narita and K.-H. Glassmeier: Spatial aliasing 3041
middle). The measured spectrum is then finally corrected forthe aliases (Fig.12 right).
Our quantitative analysis suggests that anisotropy inducedby spatial aliasing is small, say a few %, near the spec-tral peak under 20% of the longer reciprocal vector mag-nitude |k1|, except for a small ratio between the two re-ciprocal vectors|k2/k1| < 0.5 or a small angle of the twovectorsθ < 30◦. Wave telescope technique developed formulti-spacecraft measurements (Pincon and Lefeuvre, 1991;Motschmann et al., 1996; Glassmeier et al., 2001) providesthe wave vector resolution better than 1% of the reciprocalvector (see Fig.1 ranging over two orders of magnitude ofwave number) and such a wave analysis technique is capa-ble of resolving the spectrum near its peak for correctionof aliasing effect. Of course, there are always uncertaintiesin the correction procedure, in guessing the true spectrum,for example, if it is isotropic or anisotropic, or power-law orGaussian curve, or to which order the aliases are taken intoaccount (how many alias partners to be counted).
5 Conclusions
Aliasing is a general problem whenever discrete measure-ments sample continuous quantities. Periodic sampling ofa continuous function results in a periodic structure in theFourier domain. In this paper we have generalized alias-ing in the frequency domain to the wave vector domain, andshown that this spatial aliasing not only distorts the true spec-trum but also enters directly in the Brillouin zone. Aliasingin the wave vector domain has two effects, as investigatednumerically in Sect. 3. One is weak aliasing, which yieldsanisotropy and asymmetry in the distribution. The distor-tion represents primarily elongation of the distribution in theshortest lattice vector direction. Weak aliasing is determinedby the configuration of the reciprocal vectors and also by thepeak position of the energy distribution in the Brillouin zone.Another effect is strong aliasing, in which the spectral en-ergy becomes dominated by the alias in some parts of theBrillouin zone. The distribution exhibits several peaks (orig-inal peak and aliased peak) in the case of strong aliasing. Apossible method for correcting the spectrum for aliasing isalso discussed. Using several assumptions the true spectrumcan be recovered from the measured spectrum by estimatingthe aliasing contribution in the Brillouin zone. We concludeour study with a conjecture that if the correction method isapplied to the wave telescope technique for real spacecraftdata, spatial structures of the solar wind, the bow shock, andthe magnetosphere will be more precisely determined.
Acknowledgements.This work was financially supported by Bun-desministerium fur Wirtschaft und Technologie and Deutsches Zen-trum fur Luft- und Raumfahrt, Germany, under contract 50 OC0901. Discussions with M. L. Goldstein and F. Sahraoui are grate-fully acknowledged.
Topical Editor I. A. Daglis thanks J. Vogt and another anony-mous referee for their help in evaluating this paper.
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